Work on proving the rest of the "Scorpling Flugs" theorems.
Work on the Problems on logic handout.
Read Chapter 1.
Do the Chapter 1 Review Exercises.
If you want to work ahead, you can start in on Chapter 1 Exercises 1,5,8,12.
Work on the More problems on logic
handout.
Work on the Problems on sets handout.
Work on Chapter 1 Exercises 1,5,12,13. Note that the instructions for
Problem 1 are on the preceding page of the text.
Read Chapter 2 up to the subsection "Incidence Geometry." Make connections
between the text and what we have done in class on logic.
Finish the More problems on logic
handout.
Finish the Problems on sets handout.
Finish Chapter 1 Exercises 1,5,12,13.
Read Chapter 2 up to the subsection "Projective and Affine Planes."
Do the Chapter 2 Review Exercises.
Work on Chapter 2 Exercises 3-9,11,12.
Finish reading Chapter 2
Work on the Chapter 2 Exercises.
Start working on Chapter 2 Major Exercises.
Work on Chapter 2 Major Exercises.
Work on Chapter 2 Major Exercises.
Redo any returned problems that do not have a score.
Review Chapters 1 and 2 to prepare questions for the next class session.
Prepare for Friday's exam.
Submit any work on which you want feedback either by e-mail or in the box
just outside the door to the tower stairs.
Exam today.
Read Chapter 3 through the subsection "Axioms of Betweenness."
Work on Chapter 3 Exercises 1,2,6,9,12,16.
Read the handout on the Same Side and Opposite Sides
Lemmas.
Continue work on Chapter 3 Exercises 1,2,6,9,12,16.
Read all of Chapter 3 with particular attention to the subsections "Axioms
of Betweenness" and "Axioms of Congruence."
Finish Chapter 3 Exercises 1,2,6,9,12,16.
Work on Chapter 3 Exercises 24,25,26,29,32,36.
Read Chapter 3 subsections "Axioms of Continuity" and "Axiom of
Parallelism."
Read Chapter 4 through the subsection "Exterior Angle Theorem."
Finish Chapter 3 Exercises 24,25,26,29,32,36
Read Chapter 4 through the subsection "Saccheri-Legendre Theorem."
Work on Chapter 4 Exercises 9 through 14.
Finish Chapter 4 Exercises 9 through 14.
Prepare for Exam #2.
Exam #2 today
Read remainder of Chapter 4.
Work on proofs of Propositions 4.7, 4.8, 4.9, and 4.10 as assigned in
class.
Read Chapter 5.
Work on Chapter 5 Exercises 1,8,9,11.
Read Chapter 6 through the subsection "Similar Triangles".
Finish Chapter 5 Exercises 1,8,9,11.
Read Chapter 6 through the subsection "Similar Triangles".
Read Chapter 5 Exercise 4 for results to be used in Chapter 6 Exercises.
Work on Chapter 6 Exercises 2,3,4,5.
Read Chapter 6.
\Work on Chapter 6 Exercises 2,3,4,5,7(a,b),14,15.
Read Chapter 7 as outlined below.
Finish Chapter 6 Exercises 2,3,4,5,7(a,b),14,15.
Work on Chapter 7 Exercises 2,3,5,10,11.
Do Chapter 6 Review Exercises (all) and Chapter 7 Review Exercises 1-8.
Here's what to focus on in Chapter 7:
| Subsection | Pages | What to do |
|---|---|---|
| Consistency of Hyperbolic Geometry | 223-227 | Read all. |
| The Beltrami-Klein Model | 227-232 | Read all. |
| The Poincare Models | 232-238 | Read all (but just skim the paragraph on p. 237 describing the Poincare half-plane model). |
| Perpendicularity in the Beltrami-Klein Model | 238-241 | Read all. |
| A Model of the Hyperbolic Plane from Physics | 241-243 | Skip. |
| Inversion in Circles | 243-257 | Read definition of inverse on p. 243 and skim Propositions 7.1 to
7.5. Read the paragraph on p. 247 following the proof of Proposition 7.5 and relate this to the straightedge/compass construction we did in class on Monday. Read p. 248-249 on the definition of length in the Poincare disk model. Skip the rest of the section. |
| The Projective Nature of the Beltrami-Klein Model | 258-270 | Skip. |
| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| Scorpling Flugs | Theorems 1-5 | Theorems 4,5 | Thursday, May 20 |
| Problems on logic | #1-7 | #1-7 | Thursday, May 20 |
| More problems on logic | #1-4 | #4 | Friday, May 21 |
| Problems on sets | #1-6 | #4,6 | Friday, May 21 |
| Chapter 1 | RE: all E: 1,5,12,13 |
||
| Chapter 2 | RE: all E: 3-9,11,12 ME: 1,2,6,7,8 |
E: 6 (Prop 2.4 and 2.5),12 ME: 6,8 |
Wednesday, May 26 Friday, May 28 |
| Chapter 3 |
RE: all E: 1,2,6,9,12,16 E: 24,25,26,29,32,36 |
E: 9,12 E: 29,32 |
Monday, June 7 Tuesday, June 8 |
| Chapter 4 |
RE: all E: 9-14 E: 4-7 |
E: 6 or 7 |
Friday, June 18 |
| Chapter 5 |
RE: all E: 1,8,9,11 |
E: 8 |
Monday, June 21 |
| Chapter 6 |
RE: all E: 2,3,4,5,7(a,b),14,15 |
E: 14 |
Thursday, June 24 |
| Chapter 7 |
RE: 1-8 K-E: 2,3,5 P-E: 10,11 |
Course Overview and Text
This course examines three themes: the axiomatic method, specific axiom systems for geometries (both Euclidean and non-Euclidean), and the history of Euclidean and non-Euclidean geometry. A major goal for the course is learning to construct valid proofs within the specific axiom systems we study. Upon successfully completing this course, a student should be able to
The prerequisite for this course is Math 122. The main rationale for this prerequisite is to ensure that you have a certain level of mathematical experience rather than understanding of specific mathematical concepts.
Coursework and Policies
I will assign daily reading and homework. We will cover the big ideas in class and you will learn many of the details through careful reading of the text. This is particularly true of the historical context.
Most homework problems consist of constructing and writing proofs. In evaluating homework, I will look both at the validity of your reasoning and the style of your writing. If your reasoning is not valid, I might return the problem for you to try again and resubmit. My expectations for writing style will increase as the course progresses.
We will have three exams during the course. The exams are scheduled for every other Friday: May 28, June 11, and June 25.
To determine course grades, I calculate a total course score with homework problems weighted at 40% and exams weighted at 60%. I assign a preliminary course grade based on an objective standard (ususally 93.0-100% for an A, 90.0-92.9% for an A-, 87.0-89.9% for a B+, 83.0-86.9% for a B, etc.). I then look at each student's performance subjectively. Occasionally I will assign a course grade that is higher than the objective standard. For example, if a student has a grade of B according to the objective standard but has shown steady improvement, I might assign a course grade of B+.